When does infinitesimal notation break down?

Everything I've encountered in physics so far allows infinitesimal numbers to be manipulated as real numbers. But there has been much criticism towards Leibniz's notation, and I assume it is for a reason. When in mathematics will the infinitesimal notation not work? Including treating [itex]\frac{dy}{dx}[/itex] as a fraction to solve differential equations and such. Does it ever breakdown? If so, are those purely mathematical, scholarly problems, or is there places where the notation wont work in physics?

When you use dy/dx as a fraction in separating variables, you are not really doing fraction arithmetic. You are using the chain rule, and the separation of dy and dx is a shortcut way of writing that. To see that, let's consider the example dy/dx = y. Using separation of variables you would write dy/y= dx. Then you integrate both sides, the left with respect to y and the right with respect to x (also a rather suspicious maneuver) and you get log y = x + c or [tex]y = Ae^x[/tex] where A is some constant. This is a handy way to get the right answer, but here is what you are really doing: